Catalan pseudoprime

In mathematics, a Catalan pseudoprime is an odd composite number n satisfying the congruence

$(-1)^{\frac{n-1}{2}} \cdot C_{\frac{n-1}{2}} \equiv 2 \pmod n,$

where C_m denotes the m-th Catalan number. The congruence also holds for every odd prime number n that justifies the name pseudoprimes for composite numbers n satisfying it.

Properties

The only known Catalan pseudoprimes are: 5907, 1194649, and 12327121 (sequence A163209 in OEIS) with the latter two being squares of Wieferich primes. In general, if p is a Wieferich prime, then p² is a Catalan pseudoprime.

References

• Aebi, C.; Cairns, G. (2008). "Catalan numbers, primes and twin primes" (PDF). Elemente der Mathematik 63 (4): 153–164. doi:10.4171/EM/103.
• Catalan pseudoprimes. Research in Scientific Computing in Undergraduate Education.